If the population of country double in 50 years, in how many years will it triple under the assumption that the rate of increase is proportional to the number of inhabitants.

Explanation

Let x denote the population at time t in years.
Then `(dx)/(dt) propto x rArr (dx)/(dt) = kx`, Where k is constant of proportionality.
Solving `(dx)/(dt) = kx`, we get `int(dx)/x= intkdt`
or `logx=kt + c` or `x=e^(kt+c)` or `x=x_(0)e^(kt)`,
Where `x_(0)` is the population at time t=0.
Since it doubles in 50 years, at t=50, we must have `x=2x_(0)`
Hence, `2x_(0)=x_(0)e^(50k)` or `50k = log2`
or `k=(log2)/(50)` so that `50k =log2`
or `k=(log2)/(50)` so that `x=x_(0)e^((log2)/50)t`
or `t=(50 log3)/(log 2)=50 log_(2)(3)`